Fig. 1: Concept of FQFE.
Schematics of a FE, b QFE and c FQFE, supposing the ion with ±1 charges. The green and red balls represent movable ions and ligand ions, respectively. Top and side views of the FQFE example: d low-symmetry phase L1, e high-symmetry phase H, f low-symmetry phase L2. The black border indicates the unit cell. F includes two are fixed atoms (layers) F, and M is a movable atom (layer). The blue and red arrows depict the atomic displacements of M from H to L1 and L2, respectively. The green arrows show ∆P, the atomic displacements of M between L1 and L2. Since only M moves, the blue, red, and green arrows can also represent P1(polarization of L1), P2(polarization of L2), and ∆P (polarization difference between low symmetry phases), respectively. ∆P cannot be invariant under a point symmetry operation (C3z) of the low-symmetry phase, which leads to the FQFE. g The latticed form of P1, P2 and ∆P. The black dashed parallelogram depicts the “lattice” of polarization. The blue and red points represent P1 and P2, respectively. ∆P can be any vector between points with different color. Therefore, ∆P is non-zero and fractionally quantized, i.e. \(\frac{1}{3}{{{{\bf{Q}}}}}\) along the [120] direction.
